Yes, the Commissioner should be allowed to keep his job. There are a lot of tough decisions that need to be made when one is in a position of authority such as the NFL commissioner, and he did the best that he could under the circumstances. He should be allowed a second chance to prove his decision making abilities.
Roger Goodell may have made a few mistakes but that doesn't mean he should lose his job. He handled things poorly but that doesn't mean he still isn't the best man to run the NFL. If you put someone else in his place, they will be a puppet because the NFL doesn't want any more bad press. Roger Goodell still has the best interest of the league on his mind and will do whatever needs to be done.
Some argue that because the NFL is a business making record profits, Roger Goodell is immune to the controversies surrounding his reign as commisioner. Afterall, profits are the object of any business. This thinking fails to realize the irreparable damage done to the reputation of the NFL will inevitably cause a decline in profits. While fans today are willing to overlook the controversy to tune into games every week, it's likely that years down the line, the fan base won't be nearly the size it is now because of the countless scandals that plagued the NFL while Roger Goodell was running the league.
No, NFL Commissioner Roger Goodell should not keep his job, because he has not done enough to come down hard on players who violate the law. Goodell has perpetuated a culture wherein players in the NFL think that they are above the law and that they can abuse other people. Goodell is responsible for this.